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RADIO SCIENCE lournal of Research NBS/USNC-URSI 
Vol. 68D ; No. 12, December 1964 

Wave Hop Theory of Long Distance Propagation 
of Low-Frequency Radio Waves 1 

Leslie A. Berry 

Contribution From the Central Radio Propagation Laboratory, National Bureau of Standards, 

Boulder, Colo. 

(Received July 15, 1964) 

The idea that long radio waves propagating between the earth and ionosphere via 
discrete hops can be extended far into the shadow region by evaluating a series of complex 
integrals is exploited in this paper. Illustrative calculations of LF and VLF hops and total 
fields are shown as a function of distance. The second and higher hops show a pseudo- 
Brewster angle just before the caustic and attenuate like ground waves in the shadow region. 

The form of the series of integrals for an anisotropic ionosphere is given, and a model 
anisotropic ionosphere varying with height is used in a sample calculation. 

Only a small error is caused by writing each hop as the product of an effective ionospheric 
reflection coefficient and an integral which is a function only of the other characteristics of 
the path. 

1. Introduction 

Tn low-frequency radio propagation theory, it seems natural to consider the field at a 
point as the sum of rays or hops: the direct, or groundwave, plus the ray that lias been reflected 
once from the ionosphere, plus the ray that has been reflected twice from the ionosphere (and 
once from the ground), etc. [see, for example, Jollier, 1962]. This point of view is especially 
convenient for the propagation of pulses, since the different rays arrive at the receiver at 
different times because of the different length paths they have traveled. The hops are some- 
times called time-modes to indicate this separation in time. 

More precisely, for a vertical electric dipole source, the vertical electric field is written 
(with time factor exp iut suppressed) [Wait and Murphy, 1957; Johler, 19621: 

00 

E r ocE Q +J^ exp (-ihD^Dj'ajFjCj, (1) 

where E is the groundwave, Dj is the length of the ray path of the jth hop, aj is a convergence- 
divergence coefficient which corrects for focusing at the ionosphere and defocusing at the earth, 
Fj accounts for the presence of the earth at the receiver and transmitter, and Q is the effective 
reflection coefficient. For homogeneous isotropic plane earth and ionosphere [Wait and 
Murphy, 1957], 

C^RFTL, (2) 

where E e and T ee are Fresnel reflection coefficients at the ground and ionosphere, respectively, 
and 

F^{l+R e y. (3) 

The form of Cj is more complicated if the ionosphere is anisotropic [Johler, 1962], and Fj must 
be modified near grazing incidence on the earth for a spherical model [Wait and Murphy, 1957; 
Johler, 1962]. Johler [1964] indicates that (1) is adequate within about 2500 km of the source 
if F\ is modified near the caustic and in the shadow region. 



1 This work was begun as part of Advanced Research Projects Agency (ARPA) Project No. 85411 and completed on NBS Project No. 85111. 

1275 



Bremmer [1949] showed that the geometric-optic series, (1), could be obtained from the 
full-wave solution by using the saddle point approximation for certain integrals. A complete 
and very clear derivation of this connection was given by Wait [1961], who suggested that the 
field can be found at great distances with a series analogous to (1) if the integrals are evaluated 
numerically. In section 4, this series of integrals is derived for an inhomogeneous, anisotropic 
ionosphere. The derivation of the solution for the homogeneous isotropic case is outlined first 
to give the reader insight into the problem. Some sample calculations presented in section 5 
illustrate the behavior of the hops. 




Surface c 
the Earn, 



Figure 1. Geometry used. 

2. Homogeneous Isotropic Ionosphere 

The geometry is shown in figure 1 [Johler, 1962]. The center of the earth is the origin of 
a spherical (r, 6, <£) coordinate system. The surface of the earth is r=a, and the bottom of the 
ionosphere is r=g=a-\-h. An elementary vertical dipole source is located at S(b, 0, <t>), a<Cb<Cg, 
and the field is to be found at 0(r, 6, <j>), a<^r<Cg. The paths taken by the first two hops are 
shown. The angle of incidence of the jth hop on the earth is r h and <t>i, j is its angle of incidence 

on the ionosphere. The region r,<^ is the lit region for the jih hop, tj= - (grazing incidence 

on^the earth) is the caustic, and beyond the caustic is the shadow region. 
The media are characterized by their wave numbers. In the air, 

*i=7 flu (4) 

where w=27r/ is the angular frequency, c is the speed of light, and *7i is the index of refraction 
of air. In the earth, 

j CO / . At C 2 tr 2 

fc 2= _y £2 _,__, (5) 

where e 2 and a 2 are permittivity and conductivity of the earth. In the ionosphere, 



■hH* w 



2 

where cc r = . N > > co^ is the angular electron plasma frequency, and v is the collision frequency. 
The vertical electric field is (assuming r=a=b) [Johler and Berry, 1964] 

=X /w(i+ ^i^k^ (8) 

1276 



where 



£=£ hi, (9) 



l l is the dipole current moment, C is a contour in the fourth quadrant enclosing the poles of 

VirX 
— H^ K) (x), and 

H$ K) (x) is a Hankel function, 
wliere 



(2) 
(2) i a 



'(-D-VCs)- 1 - (1I) 



.(2)/ ^lfp-^Wf/) y. 



s- (2) ' '"""-^ vo.v/ <- (: 

± ee\ v ) r. c. (2 )/ //„ „\ v(2) 7 V 1 ^ 



;.(!)/ &1 f fr - Ml Wff) ..(I) fg 



where 

i?e and Tg e are called the spherical reflection coefficients since they reduce to the planar 
Fresnel reflection coefficients when the Debye approximations [Wait, I960] 



fru*r 



"VKO' 



(-i)" +1 ;ji_m d3) 



2? 
are used and - is identified as the sine of the angle of incidence: 
x 

sin 0,-7— / (14) 

and 

sin r~i — (15) 

Finally, 

>(D>(2) 

if 

brj a Bj|<i (17) 

on the contour C, then 1/(1— pR s e T s ee ) can be expanded in a geometric series. Bremmer 
[1949] and Wait [1961] assumed that (17) held on some suitable contour, and this has been 
verified numerically for all cases shown in this paper. Figure 2 shows contours of \pR s e T s ee \ 
in the complex z?-plane for a frequency of 100 kc/s and a particular model of the ionosphere. 
\p T s e Tte\ = l on the line of l's and is less than one above it. So, write 



(16) 



1+pTl 



£j*r=(l +pT' ee )(l+± P '(B'eT s ee y) (18) 

e 1 ee \ j=\ / 



1-pBi 

■=l + (l + fil) S {R'eV-'ipTQl. (19) 

Substitute (19) into (8) and integrate term by term: 

E r ^ f f(v)(l+R s e )dv+± f miX+B a e)KB%y-\pT% e ydv. (20) 

Jc }=i Jc 

1277 



Re(v) 



13200. 
i i i i i i i i i i i i i i i i i i i i i i 



-20. 



I 
m 

(v) 



-40. 



-60. 



-80. 



13400. 13600. 

i i i i i i i i i i i i i I I i i i I i i i i i i i i i 

\J1U1 !-»►-►— 



Ol 



Ol 



1 J1 \J1 v/lVJi v/l ►— ^ -C^ X- ^ ^ 

\»n ui \ji vji »— i— • -F- xUJ -h 

h- r- r- -F- -f vJ<*~ U> -^ -> 



^-> 



uJ 









i\j 



ii 



II ii 
ii 



INJ UJ I— U1M 

!\jf\j ui •— i— 

INJ U> MI- 
IS; OJOo-C* 

rsjfsj Uz-t*^ 



N 



f\J 



II II II 
II * ll II 
II II II 

II XX II 

II X X II II 

X XX II II 

X XX II 

X X X II 

X X II II 

X I I XX II 

X I I X II II 

X I I I XX II 

I I I X II 



|\J VjJ > i— 

INJ U -P* ►— I" 

N) W •> t - r~ 

KJ.vjoj ^ h- Ml- 

!\J yjO <J^ -T" »— I— 

rsjrou'^t— •— 

,-\j oj -^ >— »— >■ 



* * 



-100. + 



I x ii ii fNjy-oJ ►"*»-* • *— 

I I XX II N> oj^ »— 

I X II r\> oj^ m •- 

r\j ii x I * T * * I xx il roiNjoJ^ i— ■ ■— 

n x I * * * I I x ii ii n>\jj->^ r- »— 

I * * I ,X II fNJU/ -P*M« M 

I * * I I X II N>Lw ^l— ' I— 

I * * * I XX II rsjwoj-F*^- M 

I * • • * l x li ii ro vjj<— m »- 

I * • • • * * | | x H rvN)OJ»— ►- m 

I * • • * I XXII NJ'^H'JIH i- 

ii x I * • • * I x ll rou»»— vjnr— i— 

II X I * • • • # * I X II \)U)mi- •— 
II X I * • + • * I I X II II .^OsP^^ I'- 
ll I I I i I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 



II 



II 



X 
X 



Figure 2. Contours of equal |pT* e R||. 
The symbols, in order of decreasing magnitude, are -f , ., X, =, 2, 3, 4, 1, 5, with \pT s ee R* e = l for symbol 1. 

The first integral in (20) is exactly the groundwave [Bremmer, 1949]. If the asymptotic 



form, 



Pv-vA— cos (9) 



COS Vw 



^V^+tLm* exp [-^ + £0' 



(21) 



for #>>1, and not near or t, and the Debye approximations, (13), are used, the integrals 
in the series in (20) can be evaluated with the saddle point approximation. For more details 
of the evaluation and a discussion of the physical interpretation of the result, see Wait [1961]. 
The conclusion is that thejth term of the series in (20) is identified with thejth hop; (l-\-R s e ) 2 
corresponds to Fj in (1); {R s e ) j ~ l (T s ee ) j corresponds to C h (2); and the other factors in (20) 
supply the rest of each term in (1). The integrals in the series will be called wave hops since 
they are full wave solutions that reduce to the ray hops in the limit. 

The restrictions on the use of the Debye approximations and the location of the saddle 

point indicate that the saddle point approximation is inadequate near the caustics ( r,^- V 

Since this paper is primarily concerned with propagation to great distances into the shadow 
region, the series will be left in the integral form, (20). 

The variable of integration v has been related to the angle of incidence fa in (14), so the 
integration is essentially over the angle of incidence. Most of the contribution to the jth 
integral comes from a small part of the contour corresponding to the geometric angle of inci- 
dence. Over this part of the contour, T s ee (v) is a slowly varying function, and the Debye 



1278 



approximations for £^\ A (kig) are adequate. So, to a good approximation, T s ee (v) can be 
replaced with the constant Fresnel T ee , which is then written outside the integral [Wait, 1961]; 



E r , ^TLjjmi+RtfiRiy-ydvt* TiJj. 



(22) 



Each wave hop is the product of two factors: the ionospheric reflection coefficient, which is 
the most variable factor of a propagation path; and the path integral, I h which is a function 
of the relatively constant path parameters: ground conductivity, distance, earth curvature, 
and reflection height. Presumably, reflection coefficients for more sophisticated (e.g., con- 
tinuously varying) ionospheric models could be used in (22); indeed it is shown in the next 
section that the solution for the anisotropic case has the same path integral. Tables of the 
path integrals, if available, could be used with the published tables and graphs of reflection 
coefficients [Johler, Walters, and Harper, 1960; Wait, 1962; Wait and Walters, 1963a, b, and c; 
Walters and Wait, 1963] to calculate fields with relatively little effort. A few such results 
were given by Wait and Conda [1961]. 

For simplicity, it has been assumed that r=b=a, above. If the receiver and/or trans- 
mitter are elevated, the integrand of Ij is multiplied by 6 r , ; _^(/: 1 r)G ? ,_^(Z: 1 6), where 



v[) { ^ le) U^(M + f* (2) (^) he f 



is a height gain function. The integrand of the ground wave is multiplied by 

±1 

i a 



2) G p __j(&ir),if r<6, 



and 






m G,-,{hb)/rtr>b. 



(23) 

(24) 
(25) 



3. Homogeneous Anisotropic Ionosphere 

Now, the effect of the earth's magnetic field is considered. Assuming that 577=0, and 

o<f> 



the ionosphere is locally planar, 



where 



E T Btff(v)Q- 



+m ■ 



R" = — 



{ (1 -pT mm Rp ( 1 +pTJ +p 2 T em T me R'm }dv, 
(1 -pR\TJ (1 -pR> m T mm ) -fT em T me R e R' m } 



s a 



r " ki\ V 2j^ a 



(26) 



(27) 



The planar anisotropic ionospheric reflection coefficients T ee , T em , T mej and T mm [Johler 
and Walters, 1960] correspond to \\R\\, \\R± } ±R\\, and ±R±, respectively, in the notation of 
Wait [1962] and others. Explicit expressions for the T's are given by Johler and Berry [1964]. 

Equation (26) was derived by Wait [1963], using impedance boundary conditions, and 
by Johler and Berry [1964], using other approximate boundary conditions. 

To write (26) more concisely, define the matrices 



[Re 







T= 



T 

-*- ei 

T 

-*- mi 



Then 



E r ^jj{v)(l+R- e ) -j 



, and p=p 

I+ P T\dv 
P(R<T\' 





-R: 



(28) 



(29) 



1279 



which looks much like (8). (/ is the identity matrix.) If some norm, 
found such that 

Wp(keT\\<l 

on the contour, then \I— p(J{ e T\~ l can be expanded [Finkbeiner, 1960], 



\i-p(R.t\- 1 =\i+ji (p(R e Ty\=\i+j: gii(pT)% 

j=l j=l 



so that 



Let 



Then, since 



\I-pT\ 



M- (ReT\ 



\I- P [ReTni+pT\ = \I+(I+CR e )^Cfli^( p Ty\. 



j=i 



7j 


Xj 


-Vi 


Zj- 


1+B'e 


0" 





0_ 



{ P Ty=pi 



(i+<R.)= 






of pJleT can be 
(30) 



(3D 
(32) 
(33) 



(34) 



(35) 



Substitute (35) into (29) and integrate term by term (assuming integration and summation 
can be interchanged) : 



e t ^ f /(»)(i+#)<fo+ib f Mii+RinRiy-yy/iv. 

Jc j=l J c 



(36) 



The first integral in (36) is again the groundwave, and the second integral is the same as 
in (20) except that T j ee has been replaced by y jy the effective reflection coefficient. Indeed, if 
the magnetic field is zero, T em =T me =0, and yj=T J ee . Using (33), 



and 



71=-* ee, 

J?s-. 7?« T 72 J?s J?s fj 1 rp 

- Ll el2 ■ Li, e- L ee - Li, e J - k 'm J - em-*- me) 

\^e) 73 \^e) -*-ee ^\-^e) -^m-*- ee-*- em-*- me\ {.-E^e) \^m) ■*- em •*■ me -*■ m 



(37) 



These are the effective reflection coefficients Cj used by Johler [1961] in the geometric 
series for anisotropic ionosphere. Equation (33) gives a numerically more satisfactory method 
of computing Cj than the usual formula [Johler, 1961] : 



°' jlR.dx'll- 



-L J^mJ- mm^ 



\lt e J ee ~\~ lt m A mm )X~\ ±l e ±l m \J. ee l mm ■*- em-*- me)^ 



I 



(38) 



If the saddle point approximation is used for the integrals in (36), the geometric-optics 
series is obtained with the same limitations at great ranges as before. As in the isotropic 
case, jj can be regarded as a constant (with respect to v) so the wave hops are written 7^, 
and height gain factors (23, 24, and 25) can be included in the integrand. 



4. Inhomogeneous Anisotropic Ionosphere 

In the geometric-optic series, it is sometimes assumed that the ionosphere varies along 
the path and the reflection coefficients are calculated for the local values of the electromagnetic 



1280 



parameters. This can be done formally in (36) by writing (compare with Johler [1961]) 

E r .r= f /(tO(i+tf;. )(i+fl;,.) n 1 31 *.* n Pk T k dv, (39) 

where R s e<0 is computed for the ground parameters at the observer, R s e>s for the parameters 
at the source, and R* e)k and T k are computed for the values at the place where the ^'th hop is 
incident on the boundary for the &th time. This is an approximate formula which should be 
useful when the variation in the path is slow and smooth. Calculations using (39) will be 
made and reported in the future. 

5. Calculations and Discussion 

The integrals in (20) and (36) can be evaluated numerically using existing subroutines for 
the Hankel functions [Berry, 1964]. Figures 3, 4, and 5 show the amplitudes of the total field, 
the groundwave, and the first three wave hops as a function of distance, d=ad, for 100 kc/s, 
30 kc/s, and 10 kc/s. Propagation is over land and for the given model of the sharply bounded 
ionosphere. The curves are normalized to I l=l, (9). The caustic (grazing incidence on 
earth) is marked on each hop. One interesting feature of the second hop in figure 3 is the 
relative minimum at 3100 km — just before the caustic at 3700 km. This minimum is the 
pseudo-Brewster angle [Bremmer, 1949] of the spherical ground reflection coefficient, R s ej (10). 
In the third wave hop, R s e is squared, so t lie minimum in it is sharper. The minima occur at 
different distances in figures 4 and 5, and they are not as deep, because the pseudo-Brewster 
angle is frequency dependent. Figure 6 shows propagation of 100 kc/s waves over sea water, 
and the good conductivity nearly eliminates the pseudo-Brewster angle. 

At 10 kc/s, figure 5, the minimum at 2200 kin in ( he second hop is reflected in the third hop. 
This is hard to explain with ray theory and shows that the identification of wave hops with 
rays is not complete. 

The first hop does not show the pseudo-Brewster angle because it is not reflected at the 
ground, but propagates into the shadow region very much like a groundwave. This has been 
observed experimentally (for example, see Belrose [1964]), and was demonstrated theoretically 
by Wait [1961]. Wait and Conda [195S] showed how to calculate the effect for the first hop 
by modifying F\ 7 (1) and (3), with a "cutback" factor. All the wave hops include this effect 
implicitly. 



h l(T 9 



io-", 



i 

\ 








lEfjl, VOLTS/METER 
100 kc/s 


\Vj^5 








a =0.005, € 2 =I5 
h = 67.5 km, N=56 ELECTRONS / cm 3 


:\|E 


.ol- 




NX 


Vl.S (IO 7 ) 
TOTAL |E r | 


|Er, 2 l 




|E r . 


p^~\ 


Jt\ ^X 
r \ x^ 

J \|E r ,,l N^ 


\Y\\> 



3000 4000 5000 

DISTANCE, KILOMETERS 




3000 4000 5000 

DISTANCE, KILOMETERS 



Figure 3. Amplitude of the vertical electric field, 
and of the individual wave hops, isotropic iono- 
spheie, f=100 kc/s, land. 



Figure 4. Amplitude of the vertical electric field, 
and of the individual wave hops, isotropic iono- 
sphere, i=80 kc/s, land. 



1281 





1 








lEr.j 


.VOLTS /METER 




: 


- 










a 


10 kc/s 
0.005,e 2 =l5 






^ 


-\ 






h 


67.5 km 


N=56 ELECTRONS /cm 3 
r/ = 1.6 (I0 7 ) 




-I 




\J E r.il 


\' Er 


zl 


TOTAL |E r | 




1 














^x^ ^^\T 


- 


^ 






IE, 


.oh 




v lEr,s 






- 



3000 " 4000 5000 

DISTANCE, KILOMETERS 



f\ 






|E r ,j|, VOLTS/METER 






^^v 




IOO kc/s 


- 








a = 5, e 2 = 80 


- 






\\ 


h=67.5 km, N=56 ELECTRONS / cm3 
u--\ 6 (I0 7 ) 


- 




\ lE r>2 | 


\\ 


TOTAL |E r | 


: 


- 






\ ^^ 








\ lEr,3 


\ V\ 

\ x \h 

\ \ / \ 

lEr,,l\ \/ \ 


N 






lEr,ol\ 




\ 



Figure 5. Amplitude of the vertical electric field, 
and of the individual wave hops, isotropic iono- 
sphere, i—10 kc/s, land. 



1000 2000 3000 4000 5000 6000 7000 

DISTANCE, KILOMETERS 

Figure 6. Amplitude of the vertical elect) ic field, 
and of the individual wave hops, isotropic iono- 
sphere, 1 = 100 kc/s, sea. 



The total field at 30 kc/s, figure 4, shows an interference pattern. In mode theory, this 
is interpreted as interference between the first two modes [Johler and Berry, 1964]. Here, the 
minimum at 3500 km is seen to be interference between the first and second hops, and the 
minimum at 5600 km is interference between the second and third hops. 

For the model chosen, the wave hops are dominant for a considerable distance into their 
shadow regions. In contrast, geometric-optics cuts off the hop in the shadow region. 

Equations (20) and (36) were derived for a sharply bounded, homogeneous model iono- 
sphere. No mathematical justification has been given for replacing the ionospheric reflection 
coefficients in them with ones for a model that varies with height. There are obvious un- 
certainties; e.g., what is the effective reflection height (if there is one)? Even so, such replace- 
ment seems reasonable, especially for long radio waves. 

As an illustration, reflection coefficients for a stratified, planar, anisotropic ionosphere were 
calculated [Johler, 1962] for the daytime noon electron density profile shown in figure 7 [Pierce, 
1963]. These reflection coefficients, T ee , T em , T mej and T mm , were then used in (36) to calculate 
the propagation of the wave hops into the west over land. The earth's magnetic field vector 
was assumed to dip 60° with magnitude 0.5 G. Figure 8 shows the total field and the first 
three wave hops. The amplitude of the hops in the lit region shows that this model ionosphere 
reflects about as well as the model used in figures 3, 4, and 5. The greater attenuation of the 
total field is probably due to the lower assumed ionosphere height. 

The wave hop theory will be especially useful in the form 






(40) 



where 7; is the effective ionospheric reflection coefficient (33) and Ij is the path integral (22). 
To investigate the accuracy of (40), reflection coefficients were calculated for the sharply 
bounded, isotropic model ionosphere used in figures 3, 4, and 5 using the Fresnel formula 
[Bremmer, 1949], 



cos 



T =- 



cos 



^'"IV 1 "^ 11 *'-') 

< ^ + iV 1 ~(t sin<K 2 



(41) 



1282 









N(h 


PROFILE 






95 


1 


1 1 lllll[ 


1 1 1 !lll| 


1 1 1 1 III!) 1 


i i i mi| 


1 I 1 1 1 JLLL 


90 


~ 










/ ^ 


85 


^ 










~ 


80 


^ 










^ 


75 


^ 










- 


70 


^ 










^ 


65 


~ 










^ 


60 












^ 


55 


^ 










-^ 


50 












- 


45 












£ 


40 


~ , 


i i>rfiil i 


1 


i i i Mini i 


1 1 1 Mil! 


i i ■ i iTT 



I 10 I0 2 I0 3 10" I0 D 

N, e/cm 3 

Figure 7. Electron density profile used for calcula- 
tions shown in figure 8. 




3000 4000 5000 

DISTANCE, KILOMETERS 

Figure 8. Amplitude of the vettical electric field, 
and of the individual wave hops, stratified aniso- 
tropic ionosphere, 1 = 100 kc/s. 



o 10" 





i 


- 


j- ^ 




-= 


GROUND <r 2 = 0.0O5, € 2 =I5 \. 


^10 kc/s 




IONOSPHERE: h= 67.5 km + S S S . 






N=56, i/ = l.6xl0 7 ^ 

ISOTROPIC 


X.IOO kc/s 




FIELD 






+ + + HOP SUM WITH CONSTANT 






FRESNEL T ee 

i i i ; 







1000 2000 3000 4000 5000 



DISTANCE. KILOMETERS 

Figure 9. Comparison of field strength at 10 kc/s 
and 100 kc/s with calculations using (40). 



The angle of incidence, <j> hj , was calculated geometrically [Jollier, 1961]: 



sin<£^ 



a sin ^ 
2.7 



j2ag(l-cos^J+h 2 



(42) 



up to the caustic, where sin #*,/=- and was held constant at this value in the shadow region. 

Figure 9 compares the field at 10 kc/s and 100 kc/s with the answers given by (40) using 
7j=T ; ee , (41). The error is negligible for most applications, so the path integrals will be calcu- 
lated for a range of path parameters to complement the recent extensive reflection coefficient 
calculations [Jollier, Walters, and Harper, 1960; Wait, 1962; Wait and Walters, 1963 a, b, c; 
Walters and Wait, 1963]. 

6. Conclusions 

The wave hop theory of propagation extends the useful notion of geometric-optics to great 
distances — deep into the shadow region. The theory lends itself readily to physical interpreta- 
tion, especially in the propagation of low-frequency pulses. 



1283 



The sample calculations show the characteristics of the hops as a function of distance. 
The second and higher hops have a relative minimum just before their caustics caused by the 
pseudo-Brewster angle of the spherical ground reflection coefficient. In the shadow region the 
hops propagate much like a groundwave. 

Each wave hop can be written as the product of an ionospheric reflection coefficient and a 
path integral. The path integrals are the natural complements of the ionospheric reflection 
coefficients since they have the same form for each model of the ionosphere. They will be 
calculated for a range of parameters in the frequency range 10 kc/s to 100 kc/s and will be 
made available in the near future. 



The author is greatly indebted to J. K. Johler for initiating the research reported here and 
for guidance in its development. 

7. References 

Belrose, J. S. (1964), The oblique reflection of CW low-frequency radio waves from the ionosphere, Propagation 

of radio waves at frequencies below 300 kc/s, AGARDograph 74 (Pergamon Press, New York, N.Y.). 
Berry, L. A. (1964), Computation of Hankel functions, NBS Tech. Note No. 216. 
Bremmer, H. (1949), Terrestrial radio waves (Elsevier Publishing Co., New York, N.Y.). 

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Monograph No 78. 
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(Paper 68D 12-432) 



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